# Tutorial Track

## J. Cummings: Singular cardinal combinatorics

The tutorial will focus on the combinatorics of singular cardinals and their successors, with a special emphasis on compactness phenomena.

## M. Hušek: Large cardinals in general topology

Influence of some large cardinals on results in topological structures will be discussed. Mainly Ulam measurable cardinal, sequential cardinal and their "higher" concepts (measurable and submeasurable cardinals) will be considered. For topological structures we take topological spaces, uniform spaces, topological groups, topological linear spaces and their nice subcategories. Among the properties discussed are commutations of extensions with products, simplicity of some classes of structures, sequential continuity versus continuity and productivity of special classes of structures. One of the main tools is factorization of mappings defined in products. In some factorizations $\lambda$-strongly compact cardinals will be used.

## W. Kubiś: Generic structures

Generic structures are mathematical objects with rich automorphism groups, arising as limits of special sequences. These objects can also be described by the existence of a winning strategy in a natural infinite game where two players alternately choose bigger and bigger structures of the same type. The theory of generic structures originates from the work of Roland Fraı̈ssé in 1954 and for a long time it was considered to be part of model theory. On the other hand, one of the first generic structures was discovered by Urysohn: A universal ultrahomogeneous Polish metric space. As it happens, generic objects give rise to examples of Polish groups with very interesting dynamics.

The purpose of this tutorial is to present the theory of generic structures, with selected applications and examples from several areas of mathematics.

## J. Lopez-Abad: Extremely amenable automorphism groups

The goal of this mini course is to present the main points in the study of the topological dynamics of topological groups when represented as automorphism groups of approximate ultrahomogeneous metric structures. More particularly, we will discuss:

- Fraïssé and metric Fraïssé correspondence for (approximate) ultrahomogeneous structures (AuH).
- Representation of polish groups as automorphism groups $\mathrm{Aut}(\mathcal M)$ of (AuH) structures $\mathcal M$.
- The Kechris–Pestov–Todorcevic (KPT) correspondence between extremely amenable groups of automorphisms of (AuH) structures and the approximate Ramsey property.
- The computation of universal minimal flows.
- Examples, including

- $\mathrm{Aut}(\mathbb Q,<)$,
- $\mathrm{Aut}(\mathbb B)$, where $\mathbb B$ is the countable atomless boolean algebra,
- $\mathrm{Aut}(\mathbb G)$, where $\mathbb G$ is the Gurarij space,
- $\mathrm{Aut}(L_p[0,1])$.